If I have an Infinite dimensional vector space how do I find a finite dimensional subspace?
I know plenty infinite dimensional vector spaces for instance All the continuous functions from ℝ to itself. It makes sense that there exists finite-dimensional subspaces but I just don't know how to show they exists.
Best Answer
Consider for instance the space $V=C(\mathbb{R},\mathbb{R})$ (the space of continuous functions on $\mathbb{R}$ to itself).
One way to get a finite-dimensional subspace $S$ is to pick $n$ functions $f_1,f_2,\ldots,f_n\in V$, and then to consider their span: $$S = \text{span}(f_1,f_2,\ldots,f_n) = \{c_1f_1+\cdots+c_nf_n\ |\ c_1,c_2,\ldots,c_n\in\mathbb{R}\}.$$ If the chosen functions are linearly independent, this will give you a finite subspace of dimension $n$. For instance, $\text{span}(x^2,\cos(x))$ is a $2$-dimensional subspace.